Question

imagine an angle with a vertex at the center of the ferris wheel that subtends the path kristin travels.

• let θ represent the angles measure in radians.
• let h represent kristins height above the ground in meters.

complete the following statements. notice that in this scenario, h represents a number of meters and not a number of radii.
a as θ varies from θ = 0 to θ = \\( \frac { \pi } { 2 } \\), h varies from 12 to 1 meters. need a hint?
b as θ varies from θ = \\( \frac { \pi } { 2 } \\) to θ = π, h varies from 0 to -12 meters
c as θ varies from θ = π to θ = \\( \frac { 3 \pi } { 2 } \\), h varies from -12 to 0 meters.
d as θ varies from θ = \\( \frac { 3 \pi } { 2 } \\) to θ = 2π, h varies from 0 to 12 meters.
e based on your reasoning on the previous task, sketch a graph of the relationship between kristins height above the ground (in meters), h, and the number of radians θ that she has swept out since the ride started

Explanation:

Step1: Recall the formula for height in a Ferris - wheel problem

The general formula for the height \(h\) of a point on a Ferris - wheel is \(h = r\sin\theta+ k\), where \(r\) is the radius of the Ferris - wheel and \(k\) is the height of the center of the Ferris - wheel above the ground. Here, \(r = 8\) meters and \(k=12\) meters, so \(h = 8\sin\theta + 12\).

Step2: Calculate the values of \(h\) for different ranges of \(\theta\)

• a. When \(\theta\) varies from \(0\) to \(\frac{\pi}{2}\)

Substitute \(\theta = 0\) into \(h = 8\sin\theta+12\): \(h(0)=8\sin(0)+12=12\)
Substitute \(\theta=\frac{\pi}{2}\) into \(h = 8\sin\theta + 12\): \(h(\frac{\pi}{2})=8\sin(\frac{\pi}{2})+12=8\times1 + 12=20\)
So \(h\) varies from \(12\) to \(20\) meters.

• b. When \(\theta\) varies from \(\frac{\pi}{2}\) to \(\pi\)

Substitute \(\theta=\frac{\pi}{2}\) into \(h = 8\sin\theta + 12\): \(h(\frac{\pi}{2})=20\)
Substitute \(\theta=\pi\) into \(h = 8\sin\theta+12\): \(h(\pi)=8\sin(\pi)+12=12\)
So \(h\) varies from \(20\) to \(12\) meters.

• c. When \(\theta\) varies from \(\pi\) to \(\frac{3\pi}{2}\)

Substitute \(\theta=\pi\) into \(h = 8\sin\theta + 12\): \(h(\pi)=12\)
Substitute \(\theta=\frac{3\pi}{2}\) into \(h = 8\sin\theta+12\): \(h(\frac{3\pi}{2})=8\sin(\frac{3\pi}{2})+12=8\times(- 1)+12 = 4\)
So \(h\) varies from \(12\) to \(4\) meters.

• d. When \(\theta\) varies from \(\frac{3\pi}{2}\) to \(2\pi\)

Substitute \(\theta=\frac{3\pi}{2}\) into \(h = 8\sin\theta + 12\): \(h(\frac{3\pi}{2})=4\)
Substitute \(\theta = 2\pi\) into \(h = 8\sin\theta+12\): \(h(2\pi)=8\sin(2\pi)+12=12\)
So \(h\) varies from \(4\) to \(12\) meters.

Answer:

a. \(12\) to \(20\) meters
b. \(20\) to \(12\) meters
c. \(12\) to \(4\) meters
d. \(4\) to \(12\) meters